A note on the rank of a sparse random matrix

TL;DR

This paper extends previous results on the rank of sparse random matrices over GF(2) to arbitrary fields, providing asymptotic estimates and thresholds for full row rank, even with adversarial nonzero entries.

Contribution

It generalizes earlier work from GF(2) to any field, offering new asymptotic rank estimates and rank thresholds for sparse random matrices.

Findings

Asymptotic rank estimates for any field and fixed k≥3.

Threshold for full row rank over finite fields with random nonzero values.

Results hold even with adversarially chosen nonzero entries.

Abstract

Let $\mathbf{A}_{n,m;k}$ be a random $n \times m$ matrix with entries from some field $\mathbb{F}$ where there are exactly $k$ non-zero entries in each column, whose locations are chosen independently and uniformly at random from the set of all ${n \choose k}$ possibilities. In a previous paper (arXiv:1806.04988), we considered the rank of a random matrix in this model when the field is $\mathbb{F}=GF(2)$. In this note, we point out that with minimal modifications, the arguments from that paper actually allow analogous results when the field $\mathbb{F}$ is arbitrary. In particular, for any field $\mathbb{F}$ and any fixed $k\geq 3$, we determine an asymptotically correct estimate for the rank of $\mathbf{A}_{n,m;k}$ in terms of $c,n,k$ where $m=cn/k$, and $c$ is a constant. This formula works even when the values of the nonzero elements are adversarially chosen. When $\mathbb{F}$ is a finite field, we also determine the threshold for having full row rank, when the values of the nonzero elements are randomly chosen.